## Analysis, Manifolds, and Physics, Part 1This reference book, which has found wide use as a text, provides an answer to the needs of graduate physical mathematics students and their teachers. The present edition is a thorough revision of the first, including a new chapter entitled ``Connections on Principle Fibre Bundles'' which includes sections on holonomy, characteristic classes, invariant curvature integrals and problems on the geometry of gauge fields, monopoles, instantons, spin structure and spin connections. Many paragraphs have been rewritten, and examples and exercises added to ease the study of several chapters. The index includes over 130 entries. |

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### Contents

Review of Fundamental Notions of Analysis | 1 |

Mappings | 2 |

Berezinian | 3 |

Relations | 5 |

B Algebraic Structures Definitions | 6 |

Groups | 7 |

Rings | 8 |

Algebras | 9 |

Exterior Differential Systems | 229 |

nature | 235 |

Exterior differential equations | 236 |

First fundamental form induced metric | 238 |

Kitting vector fields | 239 |

Sphere 5 | 240 |

Curvature of Einstein cylinder | 244 |

Conformal system for Einstein equations | 249 |

Topology | 11 |

Virasoro representation of ifDiff Sl Ghosts BRST | 12 |

Separation | 13 |

Base | 14 |

Covering and compactness | 15 |

Connectedness | 16 |

Continuous mappings | 17 |

Multiple connectedness | 19 |

Associated topologies | 20 |

Topology related to other structures | 21 |

Metric spaces | 23 |

Cauchy sequence completeness | 25 |

Banach spaces | 26 |

normed vector spaces | 27 |

Banach spaces | 29 |

Hilbert spaces | 31 |

Introduction | 32 |

Measures | 33 |

Measure spaces | 34 |

Dirac adjoint | 36 |

Compactness in weak star topology | 40 |

Homotopy groups general properties | 42 |

Homotopy of topological groups | 46 |

Spectrum of closed and selfadjoint linear operators | 47 |

DIFFERENTIAL CALCULUS ON BANACH SPACES | 51 |

E Key Theorems in Linear Functional Analysis | 57 |

Problems and Exercises | 64 |

Holder spaces | 70 |

Noethers theorems II | 71 |

In variance of the equations of motion | 79 |

Calculus of Variations | 82 |

vector field | 83 |

Implicit Function Theorem Inverse Function Theorem | 88 |

DIFFERENT1ABLE MANIFOLDS | 91 |

Subgroups of Lie groups When are they Lie sub groups? | 92 |

CartanKilling form on the Lie algebra S of a Lie group G | 93 |

Differential Equations | 94 |

Direct and semidirect products of Lie groups and their Lie algebra | 95 |

Homomorphisms and antihomomorphisms of a Lie algebra into spaces of vector fields | 102 |

Homogeneous spaces symmetric spaces | 103 |

Examples of homogeneous spaces Stiefel and Grass man n manifolds | 108 |

Abelian representations of nonabelian groups | 110 |

Differentiable Manifolds Finite Dimensional Case | 111 |

Characters | 114 |

Lie algebras of linear groups | 115 |

B Vector Fields Tensor Fields | 117 |

Graded bundles | 118 |

INTEGRATION ON MANIFOLDS | 127 |

Obstruction to the construction of Spin and Pin bun dles StiefelWhitney classes | 134 |

Groups of Transformations | 143 |

Inequivalent spin structures | 150 |

Cohomology of groups | 155 |

Lifting a group action | 161 |

Short exact sequence Weyl Heisenberg group | 163 |

Cohomology of Lie algebras | 167 |

Problems and Exercises | 169 |

Quasilinear firstorder partial differential equation | 171 |

Exterior differential systems contributed by B Kent Harrison | 173 |

The strain tensor | 177 |

Backlund transformations for evolution equations con tributed by N H Ibragimov | 181 |

Poisson manifolds I | 184 |

The 2sphere | 190 |

B Integration | 212 |

Global properties | 222 |

Characteristic system | 250 |

Conformal transformation of nonlinear wave equations | 256 |

Invariants | 261 |

Masses of nomothetic spacetime | 262 |

Invariant geometries on the squashed seven spheres | 263 |

Harmonic maps | 274 |

Composition of maps | 281 |

Riemannian Manifolds Kahlerian Manifolds | 285 |

KaluzaKlein theories | 286 |

Kahler manifolds CalabiYau spaces | 294 |

B Linear Connections | 300 |

BIS CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE | 303 |

Gauge transformations | 305 |

Hopf fibering S3S2 | 307 |

Subbundles and reducible bundles | 308 |

Broken symmetry and bundle reduction Higgs mech anism | 310 |

The EulerPoincare characteristic | 321 |

Geodesies on a proper riemannian manifold | 327 |

Equivalent bundles | 334 |

Universal bundles Bundle classification | 335 |

Problems and Exercises | 336 |

Generalized Bianchi identity | 340 |

Geodetic motion equation of geodetic deviation | 344 |

Cocycles on the Lie algebra of a gauge group Anomalies | 349 |

Causal structures conformal spaces Weyl tensor | 350 |

Vbis Connections on a Principal Fibre Bundle | 357 |

operator | 363 |

DISTRIBUTIONS | 373 |

Sobolev embedding theorem | 377 |

Multiplication properties of Sobolev spaces | 386 |

The best possible constant for a Sobolev inequality on R n 3 contributed by H Grosse | 389 |

Characteristic Classes and Invariant Curvature Integrals | 390 |

HardyLittlewoodSobolev inequality contributed by H Grosse | 391 |

Spaces HsSR | 393 |

Spaces HSS and HsSW | 396 |

Completeness of a ball of XV s in JVf_ | 398 |

Distribution with laplacian in L2R | 399 |

Nonlinear wave equation in curved spacetime | 400 |

Problems and Exercises | 401 |

Harmonic coordinates in general relativity | 405 |

Leray theory of hyperbolic systems Temporal gauge in general relativity | 407 |

The geometry of gauge fields | 408 |

Einstein equations with sources as a hyperbolic system | 413 |

Wightman distributions and Schwinger functions contributed by C Doering | 414 |

Spin structure spinors spin connections | 415 |

Distributions | 423 |

Bounds on the number of bound states of the Schrod inger operator | 425 |

Sobolev spaces on riemannian manifolds | 428 |

Subject Index | 433 |

B Distributions | 435 |

Errata to Part I | 439 |

Sobolev Spaces and Partial Differential Equations | 486 |

Problems and Exercises | 525 |

Fourier transforms of expjt and expi | 531 |

Elementary solution of the wave equation | 538 |

B Theory of Degree LeraySchauder Theory | 556 |

Morse Theory | 567 |

Cylindrical Measures Wiener Integral | 573 |

Problems and Exercises | 589 |

A metric on the space of paths with fixed end points | 596 |

603 | |

Symbols | 611 |

617 | |

### Common terms and phrases

Banach space basis bijection Borel measure bounded called canonical Cauchy sequence chart compact set compact space compact support complex components compute condition connected continuous functions continuous mapping converges convex coordinates countable cr-field curve defined definition denoted diffeomorphism differentiable manifold differential equation distribution dual element equivalent Example Exercise exists fibre bundle finite dimensional following theorem Frechet space given Hausdorff hence Hilbert space homeomorphism identity implies inequality integral manifold interval inverse isomorphism Jacobi field Lebesgue integral Lie algebra Lie derivative Lie group linear form linear mapping linear operator locally compact matrix measurable function measure space metric space morphism neighborhood normed space open set p-form parameter principal fibre bundle Problem Proof properties prove real numbers respect satisfies scalar Show solution structure subgroup submanifold subset subspace supp tangent vector topological space topological vector space topology transformation unique values vanishing zero